embed_add

"We solve it by repeated substitutions in terms of rooted plane trees... ϕ(t) = sum over T in TTT(r|t,x) ... with Heaviside functions θ(tv − tw)"

"When user data are scarce, new users are represented as linear combinations of existing preference embeddings learned during training"

"M(τ) captures exactly the chosen low-order additive information induced by the fixed lift."

"fi = ⊕j=1^J hj(xi) … clr(fi) = … (b(xi) ⊗ b̃T)^⊤ θ"

"Proposition 3.2 (Identity). It holds that zii=0. ... Proposition 3.3 (Inverse consistency). ... Proposition 3.4 (Cycle consistency)"

"multi-frame trajectory consistency ... Lmrp = 1/M Σ distth(⇝pm, #»pm)"

"the mechanism requires the additive residual structure of Pre-LN transformers"

"A pretrained diffusion backbone produces a single coherent scene representation, while a lightweight conditional latent-to-latent head deterministically maps it to exposure-specific representations... exposure variation... corresponds to a structured, monotonic transformation of scene radiance... deterministic transformation of a shared scene latent"

"output summing... preserves learned module structure... linear combination naturally integrates these learned behaviours"

"Identification is achieved via a modified Wald ratio estimand, which corrects the bias due to the exclusion restriction violation"

"the resulting retrieval state is driven by both a direct contextual bias and a second-order retrieval-gate feedback loop"

"Green–Kubo formula α = ∫ E[R(xt)S(x0)] dt"

"functional equation π^{-s} Γ(s) ζ_k(s) = ... and theta transformation (1.7)"

"consecutive transitions are encouraged to agree with the longer transition they span... reversed transition is encouraged to cancel its forward counterpart"

"Hierarchical Kernel-Based Distillation (HKD)... relational kernel map P_lt... symmetric divergence-based alignment... hierarchical aggregation"

"Linearity: Zernike encoding on composite shape equals linear combination of individual encodings (Sec 6.2); harmonic pose field projection A = p · C with radially orthonormal windows guarantees invertibility (Eq 10, Sec 6.6)"

"a latent variational integrator: ... the DEL condition matches the incoming and outgoing discrete momenta at qk, giving the update its standard variational-integrator interpretation"

"Meff_eco = ∑p∈P (∑q∈P op,q) · mp (linear additive model over transitive closure)"

"Theorem 1. Ω3 is a homomorphism between <V1, +> and <PT, ⋆>: ∀ Φ1, Φ2 ∈ V1, Ω3(Φ1 + Φ2) = Ω3(Φ1) ⋆ Ω3(Φ2)"

"Theorem 1... F∗(X∗(t),t) = Σ ω∗k F(k)(X(k)(t),t) = Dt(X∗(t))"

"K = zz^⊤ + vv^⊤ − mm^⊤ − uu^⊤ ∈ R^{N×N} is conserved throughout learning ... gradient flow admits exact invariants of the learning dynamics."

"K_{n+1}[pϱ_n,...,pϱ_0] = p K_{n+1}[ϱ_n,...,ϱ_0] ... convex-linear maps on the base of a convex cone have a unique extension to the whole cone which is homogeneous of degree one"